The AlternatePaper curve begins as follows and the midpoints are numbered from 0,

|
9
|
--8--
| |
7 |
| |
--2-- --6--
| | |
1 3 5
| | |
*--0-- --4--

These midpoints are on fractions X=0.5,Y=0, X=1,Y=0.5, etc. For this AlternatePaperMidpoint they're turned 45 degrees and mirrored so the 0,1,2 upward diagonal becomes horizontal along the X axis, and the 2,3,4 downward diagonal becomes a vertical at X=2, extending to X=2,Y=2 at N=4.

The midpoints are distinct X,Y positions because the alternate paper curve traverses each edge only once.

The curve is self-similar in 2^level sections due to its unfolding. This can be seen in the midpoints as for example N=0 to N=16 above is the same shape as N=16 to N=32, but the latter rotated +90 degrees and numbered in reverse.

Arms

The midpoints fill an eighth of the plane and eight copies can mesh together perfectly when mirrored and rotated by 90, 180 and 270 degrees. The arms parameter can choose 1 to 8 curve arms successively advancing.

For example arms => 8 begins as follows. N=0,8,16,24,etc is the first arm, the same as the plain curve above. N=1,9,17,25 is the second, N=2,10,18,26 the third, etc.

The arm numbering doesn't correspond to the AlternatePaper, due to the rotate and reflect of the first arm. It ends up arms 0 and 1 of the AlternatePaper corresponding to arms 7 and 0 of the midpoints here, those two being a pair going horizontally corresponding to a pair in the AlternatePaper going diagonally into a quadrant.

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LICENSE

Copyright 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

Module Install Instructions

To install Math::PlanePath::AlternatePaperMidpoint, simply copy and paste either of the commands in to your terminal